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I am studying a bit granular dynamics and I have seen that two spheres of radius $R$ in contact with a contact area of radius $a$ would need an applied force $F$ on this two spheres that is nonlinear in the depth of deformation $\delta$ as it goes as:

$F \sim \delta^{3/2}$

To be honnest, I am not really interested in the full calculation as I am pretty sure I will forget it within two days and plus te full calculation probably would not give me a lot of insight on what is happening.

One way "a la de Gennes" that I have seen consist in relating the stess $\sigma$ to the vertical deformation $\epsilon$ via $\sigma = E \epsilon$.

Then people say that for spheres and if $a \ll R$ then

  • 1) $\epsilon \sim \delta /a$

  • 2) $a \sim \sqrt{\delta R}$

  • $\Rightarrow \:\sigma \sim E\sqrt{\delta/R}$

  • 3) $F = \pi a^2 \sigma \sim \pi R\delta \sqrt{\delta /R} \sim \pi E\sqrt{R}\delta^{3/2}$

This final result is pretty close to the actual one. My point is that I don't understand the physics of the first equation as I am used to $\epsilon = \delta L/L$ which would give me $\epsilon \sim \delta/R$.

Question:

Is there any way to understand physically the fact that $\epsilon \sim \delta/a$?

Thanks very much for any answer.

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